Align part of formula

Question

I would like to align to the left the first part of a formula inside a set. I tried using align, but it does not work well with the big paranthesis. Here is my code

\documentclass{report}
\usepackage{amsmath}
\DeclareMathOperator{\Spec}{Spec}
\usepackage[margin=1.5cm]{geometry}
\begin{document}
$\forall U\subset_d \Spec A,$
$$\mathcal{O}(U)=\left\{s:U\rightarrow\bigsqcup_{p\in U}A_p \mid  
\begin{matrix}  
\forall p\in U, s(p)\in A_p\\ 
\forall p\in U, \exists V\in\mathcal{V}(p), V\subset U, \exists a,f\in A, \text{ astfel \^inc\^at } \forall q\in V, f\notin q, s(q)=\frac{a}{f} \text{ \^in }A_q
\end{matrix}
\right\}$$
\end{document}`

And here is a picture of what I am trying to accomplish :

Answer 1

You can use aligned:

\documentclass{report}
\usepackage[margin=1.5cm]{geometry}
\usepackage{amsmath}

\DeclareMathOperator{\Spec}{Spec}

\begin{document}

$\forall U\subset_d \Spec A$,
\[
\mathcal{O}(U)=
\biggl\{s\colon U\rightarrow\bigsqcup_{p\in U}A_p
\;\bigg|\;
\begin{aligned}
&\forall p\in U, s(p)\in A_p\\
&\forall p\in U, \exists V\in\mathcal{V}(p), V\subset U, \exists a,f\in A,
  \text{ astfel \^inc\^at } \forall q\in V, f\notin q, s(q)=\tfrac{a}{f} \text{ \^in }A_q
\end{aligned}
\biggr\}
\]

\end{document}

Some notes.

1. Punctuation should go after inline formulas
2. \left\and \right give too big delimiters in this case
3. $$ should be never used in LaTeX
4. \tfrac is used not to make a big fraction